Question

Let \(X\) be a compact Hausdorff space. An ideal in \(C(X, \mathbb{R})\) is a subalgebra \(\mathcal{J}\) of \(C(X, \mathbb{R})\) such that if \(f \in J\) and \(g \in C(X, \mathbb{R})\) then \(f g \in J\). a. If \(J\) is an ideal in \(C(X, \mathbb{R})\), let \(h(\mathcal{J})=\\{x \in X: f(x)=0\) for all \(f \in \mathcal{J}\\} .\) Then \(h(\mathcal{J})\) is a closed subset of \(X\), called the hull of J. b. If \(E \subset X\), let \(k(E)=\\{f \in C(X, \mathbb{R}): f(x)=0\) for all \(x \in E\\}\). Then \(k(E)\) is a closed ideal in \(C(X, \mathbb{R})\), called the kernel of \(E\). c. If \(E \subset X\), then \(h(k(E))=\bar{E}\). d. If \(J\) is an ideal in \(C(X, R)\), then \(k(h(\mathcal{J}))=J\). ( Hint: \(k(h(J))\) may be identified with a subalgebra of \(C_{0}(U, \mathbb{R})\) where \(\left.U=X \backslash h(J) .\right)\) e. The closed subsets of \(X\) are in one-to-one correspondence with the closed ideals of \(C(X, \mathbb{R})\).

Short answer

a) Hull is closed. b) Kernel is a closed ideal. c) Hull of kernel is closure. d) Kernel of hull equals the ideal. e) Correspondence between closed sets and closed ideals.

Step-by-step solution

Define the Hull of an Ideal

Given an ideal \( \mathcal{J} \) in \( C(X, \mathbb{R}) \), the hull \( h(\mathcal{J}) \) consists of points \( x \in X \) where every function in \( \mathcal{J} \) vanishes. As it is the intersection of the zeros of all functions in \( \mathcal{J} \), it follows that \( h(\mathcal{J}) \) is a closed set.

Establish Hull of Kernel Relationship

For any subset \( E \subseteq X \), the kernel \( k(E) \) is the collection of functions in \( C(X, \mathbb{R}) \) that vanish on \( E \). The hull of the kernel \( h(k(E)) \) contains points where all functions in \( k(E) \) vanish, which by definition of kernel are the closure of \( E \) hence \( \bar{E} \).

Prove the Kernel of Hull of an Ideal

Given an ideal \( \mathcal{J} \), consider its hull \( h(\mathcal{J}) \), where each function in the kernel \( k(h(\mathcal{J})) \) vanishes. By the properties of compact Hausdorff spaces and ideals, \( k(h(\mathcal{J})) = \mathcal{J} \) since \( h(\mathcal{J}) \) defines precisely the vanish points of functions in \( \mathcal{J} \).

Analyze the Correspondence Between Closed Subsets and Ideals

Closed subsets of \( X \) map to closed ideals in \( C(X, \mathbb{R}) \) by associating each closed subset with a kernel \( k(F) \) representing continuous functions vanishing on \( F \). Conversely, closed ideals map to their hulls, creating a one-to-one correspondence by this kernel-hull duality.

Key concepts

Ideals in Algebra

Ideals in algebra play a fundamental role in understanding the structure of algebraic systems. An ideal in a ring is a special kind of subset that allows ring operations to be "compatible" with the subset. In the context of the exercise, we discuss ideals within the algebra of continuous real-valued functions on a compact Hausdorff space, denoted as \(C(X, \mathbb{R})\).

Here, an ideal \(\mathcal{J}\) in \(C(X, \mathbb{R})\) must be a subset such that if a function \(f\) belongs to \(\mathcal{J}\) and \(g\) is any function in \(C(X, \mathbb{R})\), then their product \(fg\) is also in the ideal \(\mathcal{J}\). This property ensures that the multiplication operation in the algebra respects the constraints imposed by the ideal.

Ideals help to define important algebraic concepts such as kernels and images, which are crucial for understanding function properties and mappings in algebraic topology and analysis.

In practical terms, ideals allow us to understand and manipulate the continuity, zeros, and other characteristics of groups of functions rather than dealing with each function individually.

Continuous Functions

Continuous functions are those functions that vary smoothly without abrupt changes in value. In mathematical terms, a function \(f: X \to \mathbb{R}\) is continuous on a space \(X\) if for every point \(x\) in \(X\) and every epsilon \(\varepsilon > 0\), there exists a delta \(\delta > 0\) such that for all points \(y\) in \(X\), if the distance between \(x\) and \(y\) is less than \(\delta\), then the difference in function values \(|f(x) - f(y)|\) is less than \(\varepsilon\).

In the setting of a compact Hausdorff space, all continuous functions on \(X\) take on bounded values, and sequences of continuous functions behave particularly nicely, converging uniformly under appropriate conditions.

Continuous functions form an algebra because they can be added, multiplied, and scaled by real numbers, while still being continuous. This is why they serve as ideal candidates for the study of algebraic structures in topological spaces, bridging the gap between algebra and topology.

In algebraic topology, understanding continuous functions on a compact Hausdorff space is essential as they dictate the behaviors and properties of those spaces significantly.

Kernel and Hull Correspondence

The kernel and hull correspondence describes a fascinating relationship in the study of continuous functions on compact Hausdorff spaces. This duality is crucial in both algebra and topology due to its ability to relate the presence of function zeros with subalgebra structures.

Let’s break it down:

• The **kernel** of a subset \(E \subset X\), denoted \(k(E)\), is the ideal of continuous functions that vanish everywhere on \(E\). This means any function in \(k(E)\) evaluated at points in \(E\) yields zero.
• The **hull** of an ideal \(\mathcal{J}\), represented as \(h(\mathcal{J})\), encompasses all points \(x\in X\) such that every function in \(\mathcal{J}\) becomes zero at that point. The hull describes the collective "zero set" of all functions contained within the ideal.

When we take the kernel of the hull of an ideal, we end up recovering the original ideal, \(k(h(\mathcal{J})) = \mathcal{J}\), showing a precise correspondence.

Moreover, the hull of the kernel \(h(k(E)) = \bar{E}\) yields the closure of \(E\), demonstrating a meaningful topological property: the way ideal structures map closed subsets and vice versa, explaining the kernel-hull duality's one-to-one correspondence nature.